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The boundary line shown is 2 x + 3 y = 6 . Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus 3 y equals 6 is plotted as a dashed arrow extending from the top left toward the bottom right. The coordinate plane to the bottom of the line is shaded.

The line 2 x + 3 y = 6 is the boundary line. On one side of the line are the points with 2 x + 3 y > 6 and on the other side of the line are the points with 2 x + 3 y < 6 .

Let’s test the point ( 0 , 0 ) and see which inequality describes its side of the boundary line.

At ( 0 , 0 ) , which inequality is true:

2 x + 3 y > 6 or 2 x + 3 y < 6 ? 2 x + 3 y > 6 2 x + 3 y < 6 2 ( 0 ) + 3 ( 0 ) > ? 6 2 ( 0 ) + 3 ( 0 ) < ? 6 0 > 6 False 0 < 6 True

So the side with ( 0 , 0 ) is the side where 2 x + 3 y < 6 .

(You may want to pick a point on the other side of the boundary line and check that 2 x + 3 y > 6 .)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality 2 x + 3 y < 6 .

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Write the inequality shown by the shaded region in the graph with the boundary line x 4 y = 8 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 4 y equals 8 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the top of the line is shaded.

x 4 y 8

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Write the inequality shown by the shaded region in the graph with the boundary line 3 x y = 6 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the right of the line is shaded.

3 x y 6

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Graph linear inequalities

Now, we’re ready to put all this together to graph linear inequalities.

How to graph linear inequalities

Graph the linear inequality     y 3 4 x 2 .

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Identify and graph the boundary line. If the inequality is less than or equal to or greater than or equal to, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. The text in the second cell reads: “Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y equals three-fourths x minus 2. The inequality sign is greater than or equal to, so we draw a solid line. The third cell contains the graph of the line three-fourths x minus 2 on a coordinate plane. In the second row of the table, the first cell says: “Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality? In the second cell, the instructions say: “We’ll test (0, 0). Is it a solution of the inequality?” The third cell asks: At (0, 0), is y greater than or equal to three-fourths x minus 2? Below that is the inequality 0 is greater than or equal to three-fourths 0 minus 2, with a question mark above the inequality symbol. Below that is the inequality 0 is greater than or equal to negative 2. Below that is: “So (0, 0) is a solution. In the third row of the table, the first cell says: “Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. In the second cell, the instructions say: The test point (0, 0) is a solution to y is greater than or equal to three-fourths x minus 2. So we shade in that side.” In the third cell is the graph of the line three-fourths x minus 2 on a coordinate plane with the region above the line shaded.
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Graph the linear inequality y 5 2 x 4 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-halves x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.

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Graph the linear inequality y < 2 3 x 5 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals two-thirds x minus 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.

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The steps we take to graph a linear inequality are summarized here.

Graph a linear inequality.

  1. Identify and graph the boundary line.
    • If the inequality is or , the boundary line is solid.
    • If the inequality is<or>, the boundary line is dashed.
  2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. Shade in one side of the boundary line.
    • If the test point is a solution, shade in the side that includes the point.
    • If the test point is not a solution, shade in the opposite side.

Graph the linear inequality x 2 y < 5 .

Solution

First we graph the boundary line x 2 y = 5 . The inequality is < so we draw a dashed line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right.

Then we test a point. We’ll use ( 0 , 0 ) again because it is easy to evaluate and it is not on the boundary line.

Is ( 0 , 0 ) a solution of x 2 y < 5 ?
The figure shows the inequality 0 minus 2 times 0 in parentheses is less than 5, with a question mark above the inequality symbol. The next line shows 0 minus 0 is less than 5, with a question mark above the inequality symbol. The third line shows 0 is less than 5.

The point ( 0 , 0 ) is a solution of x 2 y < 5 , so we shade in that side of the boundary line.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The point (0, 0) is plotted, but not labeled. The region above the line is shaded.

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Graph the linear inequality 2 x 3 y 6 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 3 y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.

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Graph the linear inequality 2 x y > 3 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 3 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.

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What if the boundary line goes through the origin? Then we won’t be able to use ( 0 , 0 ) as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Graph the linear inequality y −4 x .

Solution

First we graph the boundary line y = −4 x . It is in slope–intercept form, with m = −4 and b = 0 . The inequality is so we draw a solid line.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line s y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right.

Now, we need a test point. We can see that the point ( 1 , 0 ) is not on the boundary line.

Is ( 1 , 0 ) a solution of y −4 x ?
The figure shows 0 is less than or equal to negative 4 times 1 in parentheses, with a question mark above the inequality symbol. The next line shows 0 is not less than or equal to negative 4.

The point ( 1 , 0 ) is not a solution to y −4 x , so we shade in the opposite side of the boundary line. See [link] .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right. The point (1, 0) is plotted, but not labeled. The region to the left of the line is shaded.
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Graph the linear inequality y > −3 x .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a dashed arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.

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Graph the linear inequality y −2 x .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x is plotted as a solid arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.

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Some linear inequalities have only one variable. They may have an x but no y , or a y but no x . In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

x = a vertical line y = b horizontal line

Graph the linear inequality y > 3 .

Solution

First we graph the boundary line y = 3 . It is a horizontal line. The inequality is>so we draw a dashed line.

We test the point ( 0 , 0 ) .

y > 3 0 > 3

( 0 , 0 ) is not a solution to y > 3 .

So we shade the side that does not include (0, 0).
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.

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Questions & Answers

Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in $45 and $25 stock shares. The number of $25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy?
Jawad Reply
let # of $25 shares be (x) and # of $45 shares be (y) we start with $25x + $45y=475, right? we are told the number of $25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
A private jet can fly 1,210 miles against a 25 mph headwind in the same amount of time it can fly 1,694 miles with a 25 mph tailwind. Find the speed of the jet
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Mckenzie
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Reiley Reply
Mr Hernaez runs his car at a regular speed of 50 kph and Mr Ranola at 36 kph. They started at the same place at 5:30 am and took opposite directions. At what time were they 129 km apart?
hamzzi Reply
90 minutes
muhammad
Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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